Meshes for example 6.1.

## Abstract

Most engineering applications estimate the deformation induced by loads by using the linear elasticity theory. The discretization process starts with the equilibrium equation and then develops a displacement formulation that employs the Hooke’s law. Problems of practical interest encompass designing of large structures, buildings, subsurface deformation, etc. These applications require determining stresses to compare them with a given failure criteria. One often tackles this way a design or material strength type of problems. For instance, Geomechanics applications in the oil and gas industry assess the induced stresses changes that hydrocarbon production or the injection of fluids, i.e., artificial lift, in a reservoir produce in the surrounding rock mass. These studies often include reservoir compaction and subsidence that pose harmful and costly effects such as in wells casing, cap-rock stability, faults reactivation, and environmental issues as well. Estimating these stress-induced changes and their consequences require accurate elasticity simulations that are usually carried out through finite element (FE) simulations. Geomechanics implies that the flow in porous media simulation must be coupled with mechanics, which causes a substantial increase in CPU time and memory requirements.

### Keywords

- elasticity
- single-phase flow
- geomechanics
- Dirichlet-Neumann
- mortar methods
- continuous Galerkin

## 1. Introduction

This chapter presents a continuous Galerkin FE formulation for linear isotropic elasticity. It covers in detail how to derive such formulation beginning with the equilibrium equation and the virtual work statement. It also discretizes the continuity equation for slightly compressible single-phase flow to show how to couple different physics with elasticity. It discusses several coupling approaches such as the monolithic and iterative ones, i.e., loosely coupled. This chapter also mentions the affinity of the poroelastic case with the thermoelastic one. It thus also includes thermoelasticity in the treatment herein. It shows concrete numerical examples covering two- and three-dimensional problems of practical interest in thermo-poroelasticity. The sample problems employ triangular, quadrilateral, and hexahedral meshes and include comments about implementing boundary conditions (BCS). An introduction to domain decomposition ideas such as iterative coupling by the BCS, i.e., Dirichlet-Neumann domain decomposition and mortar methods for non-matching interfaces is included.

The treatment herein demonstrates that the continuous Galerkin formulation for linear isotropic elasticity is the foundation to develop codes for mechanics. Indeed, after discretizing linear elasticity is straightforward to extend the implementation to nonlinear mechanics such as rate-independent plasticity. It thus provides some comments about such extension. Applications of practical interest show that industrial size problems will require domain decomposition techniques to handle such simulations in a timely fashion. Unquestionably, domain decomposition techniques can exploit current parallel machines architectures which brings high-performance computing into the picture. For instance, recently the author showed that the Dirichlet-Neumann scheme could handle problems at the reservoir field-level as well as the mortar method decoupled by this last one. Its current results are backed up by papers published in peer-reviewed journals and conferences thus this book chapter summarizes that effort.

## 2. Mathematical model for thermo-poroelasticity

This section discusses the governing equations for linear homogeneous isotropic thermo-poroelasticity and their FE formulation. It skips details for the sake of brevity thus a more detailed treatment can be found in [1, 2, 3, 4]. The mathematical formulation considers a bounded domain

where the equation’s parameters are

where the additional parameters are accordingly

one should also consider an initial or reference pressure distribution in the whole domain. Sources and sinks simulate injector and producer wells, respectively. Herein

where

where

One can derive a weak form by substituting Eq. (2) into Eq. (1) and then multiplying by a test function

A weak form for the equilibrium Eq. (4) can be derived in a similar way, by testing against a given virtual displacement,

where

where

where

here

where

Finally substituting the generalized Hooke’s law Eq. (5) into Eq. (8) and using Eq. (7) leads to the FE model for linear isotropic poroelasticity, thus:

One can obtain the loose coupling approach in different ways. Eq. (15) shows one possible choice, where one solves the displacements first by taking the pressures from the previous time step. Next, one updates the pressures by using the newest displacements:

where expressions for the matrices are provided in Eq. (16) and

This section completes with a comment about the Continuous Galerkin (CG) formulation for the pressure (1). It is well-known that the formulation that was presented above for flow it is not locally mass conservative, and thus the resulting fluxes are not continuous across the element edges. But it is also true that accurate flow simulations require the latter, especially for multi-phase flow, though. Nevertheless, one can utilize post-processing techniques to recover locally conservative mass fluxes [2]. This chapter, though, for convenience has restricted its focus to CG methods for flow but has realized that the coupled formulation may be modified to consider mixed FE methods and finite volumes for flow as well as changing CG by post-processing. The author already showed for the simple flow cases reported herein that CG yields to physical pressure fields that can be employed for geomechanics purposes. The precise numerical comparison among CG and Discontinuous Galerkin (DG) solutions was performed in [2] to demonstrate that CG can compute pressures accurately.

## 3. Nonlinear heat transfer equation

The transient nonlinear heat conduction in a given body is as follows [9, 10, 11]:

In (17),

One can derive a FE formulation for model problem (17) by multiplying by a test function and integrate by parts and applying the Gauss-divergence theorem to arrive at the following bilinear form:

where the functions are:

Time discretization renders the local residual for the element

where the linear operator

this equation renders once again:

if one assumes that

where the variation term is given by:

One often employs the Newton-Raphson algorithm to solve the linearized system of equations in every time step, namely,

## 4. Domain decomposition methods

Domain Decomposition Methods (DDM) encompass highly efficient algorithms to obtain solutions of large-scale discrete problems on parallel super-computers. They mainly consist of partitioning the domain into various subdomains and then getting the global solution through the resolution of the subdomain problems [12, 13] often in an iterative fashion. These methods can be seen as an iterative coupling by the internal and thus unknown BCS. There is a broad literature covering these approaches, and that is why this chapter, therefore, presents a short introduction for the sake of completeness. The recommended references include Bjorstad and Widlund [14], Bramble et al. and Marini and Quarteroni [15], who considered the Dirichlet-Neumann (DN) DDM and Neumann-Neumann.

Let

Let the primary variable be called “displacements” and their gradient “tractions,” i.e., normal derivative in the boundary. Then, the tractions on the interface

It happens that this approach requires at least a two-entry coloring tool or even more, i.e., there may be subdomains with mixed interfaces, colored as gray [12]. There is a lack of parallelism in the sense that black subdomains must wait for the white ones to communicate their tractions. An initial guess for tractions should be prescribed to mitigate this issue, but this latter is not feasible in most cases. A straightforward way to obtain an initial estimate for the multiplier

## 5. The mortar FE method (MFEM)

The primary goal here is to extend MFEM to glue curved interfaces such as the one shown in Figure 2 where MFEM treats non-matching interfaces. The section first introduces a brief description of non-uniform rational B-Splines curves and surfaces (NURBS) in [2, 3, 18]. The reader is referred to those references that cover the topics of computational geometry, in particular how to build these NURBS entities. Let MFEM be described for linear isotropic elasticity regarding bilinear forms,

where

the parameters in Eq. (28) are as follows:

herein

One can write in a matrix or algebraic form, Eq. (28) as:

The equation above corresponds to the so-called “saddle-point problem (SPP).” Notice that subdomains are only connected using the Lagrange multiplier

The following line integral defines the projector, for 2-D problems, as:

where

where

## 6. Numerical examples

The author implemented these FE models in the Integrated Parallel Finite Element Analysis program (IPFA) that is a C++ application whose main characteristics are described in [2, 12]. IPFA employs standard continuous Lagrange polynomials as shape functions for the space discretization in each subdomain,

### 6.1. Example 1: Two-dimensional steady state single-phase flow

The example is a manufactured problem where the solution is a priori chosen. Then, one substitutes the given pressure field in the governing partial differential equation to obtain the source term, i.e., loading, that reproduces the input field. The problem in strong form looks like:

where the domain of interest corresponds to the unitary square and Dirichlet BCS are enforced. The input pressure field is given by:

Figure 3 shows the pressure field that corresponds to the problem 6.1 whose discretization encompasses three subdomains: two of them (the top and bottom ones) consist of triangular meshes while the one in the middle was discretized by a regular Cartesian quadrilateral mesh. The top-left corner of the figure shows the mesh that is employed.

The pressure field is on the right-top corner, and its horizontal derivative is in the bottom-left corner, while the discrepancy between the numerical and exact solutions, i.e., the absolute error, was rendered in the right-bottom corner. Table 1 represents the number of elements and points of each mesh from top to bottom. The mortars as geometrical entities correspond to two B-Splines interpolants (NURBS with all weights equal 1) that were constructed by interpolating a sinusoidal wave as the figure shows (see [3] for details). Thirty-two quadratic mortar elements per curve were utilized to glue these three subdomains. A direct frontal solver was used to solve the global SPP in Eq. (30) [3]. The results that are summarized on Figure 3 are in good agreement with the analytical solution. The absolute error against the correct answer is also displayed. The discrepancy is of the order of

Points | Elements | Kind of mesh |
---|---|---|

980 | 1814 | Triangular |

1560 | 1472 | Quadrilateral |

4090 | 7858 | Triangular |

Whether or not one utilizes the SPP approach, the local problems are completely disconnected. This fact can be exploited to reduce the computational time significantly. Indeed, these sub problems can be handled in separate threads using a shared memory approach, i.e., multi-threading assembling. A convergence analysis was also performed, by successively running refined meshes [3] and by keeping a refinement ratio of 2:1 between subdomains. The exercise used a piecewise quadratic mortar space where the number of mortar elements equals the number of coarse edges in the non-mortar sides. It tackled meshes of size 8, 16, 32, 64, 128 and 256 respectively. Figure 4 displays the resulting convergence rate in a

Finally, Figure 5 shows pressure snapshots that represent four different Dirichlet-Neumann iteration levels evolving from left-to-right and top-to-bottom. The fact that no initial guess for pressure was provided explains the mismatch in the first snapshot. That is why one needs to eliminate discrepancies by running the process to match up those subdomains, i.e., the traction residual in the interface must vanish, which for this case occurs in just a handful of iterations. The stop criterion precisely involves the residual in the tractions in the interface that is required to fall below the given tolerance. For this particular problem, the iterative process spent six iterations to achieve a residual lower than

### 6.2. Example 2: Coupled flow and mechanics

This example analyzes a coupled flow and mechanics simulation in a reconstructed reservoir (RS) model with different meshes for the flow and mechanics physics [18]. The author proposed such a reconstruction workflow in [18] which permits this latter feature by computing a projection operator to mapping pressures from the original flow mesh into the so-called reference mechanics mesh. Toward that end, the example employs the slightly compressible flow formulation loosely combined with the mechanics model as shown in Eq. (15). The objective is to show a realistic field level RS compaction and subsidence coupled computation. The goal is thus working three different cases for the mechanics part in which one only changes the resolution of the reconstructed mechanics mesh in the pay-zone while preserving the mechanical properties constant as well as the geometry, BCS, and the depletion scenario. The exercise admits the actual static properties as being in the pay-zone such as porosity

Table 2 compiles the mesh dimensions in every direction. The example also contemplates

Case # | Description | Nx | Ny | DOF | Assembling time | |
---|---|---|---|---|---|---|

One | 1/4 of RS | 35 | 13 | 15,960 | 51,830 | 0 min, 19 s 75 ms |

Two | 1/2 of RS | 70 | 26 | 48,279 | 159,120 | 0 min, 59 s 89 ms |

Three | 1/1 of RS | 140 | 49 | 156,408 | 506,160 | 3 min, 14 s 89 ms |

Figure 8 displays the mechanics mesh. The second case on Table 2 corresponds to a layered RS with Young’s modulus

Figure 9 pictures snapshots with the evolution of the vertical displacement

Figure 10 renders pressure-drop snapshots at 10 years of production. Each picture draws the original RS mesh and the reference mechanic’s mesh for all cases that Table 2 covered, from top-to-bottom and left-to-right. Notice that the action of the projection operator improves with the refinement of the reference mechanics mesh as one should expect. The monotone pressure behavior, which does not drastically change across neighboring elements in the original RS mesh, may explain this improvement. Though, some items remain red-colored because they are inactive. That happens due to the interpolation error that tends to smooth out the RS topology. Perhaps it is not clear in the picture, but the reference mechanics mesh’s layers (since the thickness distribution in the

Finally, Table 3 reviews results for the minimum and maximum vertical displacements

Case # | Runtime | ||
---|---|---|---|

One | −6.652 | 2.693 | 4 min, 34 s 23 ms |

Two | −6.511 | 2.961 | 7 min, 53 s 84 ms |

Three | −6.469 | 2.752 | 23 min, 42 s 18 ms |

The above-coupled flow and geomechanics computation, which used the reconstructed model, confirmed that the procedure is quite useful to tackle realistic reservoir compaction and subsidence simulations [18].

### 6.3. Example 3: Nonlinear heat transfer: arch problem

The example addresses the interesting problem that has been investigated by several researchers [9, 10]. Its distinctive features are the two re-entrant corners. Near sharp corners, there may be singularities in the solution, which cause the spatial derivatives of the solution to become unbounded. The material properties are constant density and specific heat, and a linear isotropic thermal conductivity,

Figure 12 shows the domain and the mesh. The BCS are of Dirichlet type on the left- (

which is the short-time linear solution at a time

Figure 13 shows temperature field snapshots for different times increasing from top to bottom. The example simulates 0.1 s with a fully implicit approach. It is observed that a heating front quickly travels from left to right as expected due to the temperature gradient. The temperature scale in the color maps is from 0 to 1000°K. As a qualitative benchmark, the temperature profile reported by Winget and Hughes [9] accords very well with the results herein.

The example finalizes with a simple loosely coupled thermal and mechanics computation. It takes the temperature variation that the arch problem experiences as driving force for the mechanical problem. It assumes linear isotropic elasticity with

## 7. Concluding remarks

This chapter introduced how to estimate stress-induced changes using elasticity simulations that are often performed through FE computations. It thus presented a formulation for linear thermo-poroelasticity. It covered the nonlinear energy equation as well. It also implemented a comprehensive MFEM on curved interfaces where the classical DN-DDM was employed to decouple the global SPP for elasticity, and steady single-phase flow. The coupled flow and geomechanics computation that utilizes the reconstructed model showed that this workflow is valuable to tackle realistic reservoir compaction and subsidence simulations. The research presented herein unfolds new prospects to further parallel codes for reservoir simulation coupled with geomechanics.

## Acknowledgments

The author recognizes the financial support of the project “Reduced-Order Parameter Estimation for Underbody Blasts” financed by the Army Research Laboratory, through the Army High-Performance Computing Research Center under Cooperative Agreement W911NF-07-2-0027 and also acknowledgments Dr. Belsay Borges for proofreading the manuscript.